Question

In $ \triangle ABC $, with usual notations, $ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{C}{2} \right) = \sin \left( \frac{B}{2} \right) \quad \text{and} \quad 2s \text{ is the perimeter of the triangle. Find the value of } s. $ Then the value of $ s $ is:

• A. \( 2b \)
• B. \( 6b \)
• C. \( 3b \)
• D. \( 4b \)

Hint:

When solving for sides or semi-perimeter in a triangle using trigonometric identities, use the sine rule and the given relations to find the appropriate expressions.

Answer 1

The Correct Option is C

We are given that: \[ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{C}{2} \right) = \sin \left( \frac{B}{2} \right) \] and that \( 2s \) represents the perimeter of the triangle. 
Using the sine rule and the relation between the sides and angles of the triangle, we can use the formula for the semi-perimeter of the triangle.
Based on the given conditions, solving for the perimeter and angle relations leads us to the solution: \[ s = 3b \] 
Thus, the correct answer is \( 3b \).